\(h\)-stability for linear dynamic equations on time scales
type: Item
instance of: scholarly article
title: \(h\)-stability for linear dynamic equations on time scales
zbMATH Open document ID: 1112.34031
author: Sung Kyu Choi
author: Namjip Koo
author: Dong Man Im
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: This paper extends the concept of \(h\)-stability introduced e.g., by \textit{M.~Pinto} [Appl.\ Anal.\ 43, 1--20 (1992; Zbl 0748.34029)] to dynamic equations on time scales. This stability notion is quite flexible, since it includes the classical theories of uniform or uniform exponential stability within one common framework. The authors show that \(h\)-stability of linear systems is preserved under Lyapunov transformations (cf \textit{B.\ Aulbach} and the reviewer [J.\ Comput.\ Appl.\ Math.\ 141, No. 1-2, 101--115 (2002; Zbl 1032.39008)]). A Gronwall argument yields robustness w.r.t\ linear perturbations. Using a Bihari-type inequality, similar results are also derived for nonlinear perturbations. Finally, quadratic Lyapunov functions enable the authors to obtain sufficient conditions for \(h\)-stability of linear systems. An example and various special cases illustrate the paper.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 39A14
Mathematics Subject Classification ID: 39A13
zbMATH DE Number: 5078094
zbMATH Keywords: linear dynamic equation
zbMATH Keywords: \(h\)-stability
zbMATH Keywords: quadratic Lyapunov function
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.046
OpenAlex ID: W1981311117
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DOI: 10.1016/J.JMAA.2005.12.046
description: scientific article; zbMATH DE number 5078094